You have 2 options.
1. You can disembark last with everyone else who is doing B2B, go through customs saying you have nothing to declare, and get back on the ship. It shouldn't take more than 30mins. Usually they will have a small spread with coffee/tea/pastries while you wait to go.
2. You can get off when you want and spend the day in port, visit a beach/shop/whatever. Must be back on 90 minutes prior to departure time.
As for moving cabins I'd say it depends on if you spread out a lot in the cabin or just take a shirt out of the suitcase when you need it. If you change rooms you will need to repack so the steward can move your luggage. If the cost is small I'd upgrade and stay in the same room. One less hassle to worry about.